On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles

TL;DR

This paper presents a method using Worpitzky Number Triangles to efficiently determine polynomial coefficients from sequences, applicable to sequences with various starting points and differences.

Contribution

It introduces a novel approach leveraging two specific Worpitzky Number Triangles to solve for polynomial coefficients via simple linear equations.

Findings

Method successfully applied to sequences starting at 0 and 1

Extended to sequences with arbitrary starting points and differences

Provides a straightforward lookup table approach for coefficient calculation

Abstract

In this paper we show that two of the Worpitzky Number Triangles, OEIS A028246 and A019538, may each be used in look-up table fashion, along with specific diagonals of a polynomial sequence's difference triangle to easily solve for the unknown coefficients of the sequence. This is accomplished by using a method that isolates each of the coefficients as a single unknown in a series of simple linear equations. The method is first applied to a sequence generated using integer indexes with a starting index of 0, using the Alternate Worpitzky Number Triangle, A019538. Although the numbers in A019538 are less commonly referred to as a Worpitzky Number Triangle, a justification for such a reference is briefly presented. Next, the method is applied to a sequence generated using integer indexes with a starting index of 1, using the Mirrored Worpitzky Number Triangle, A028246. The method is then extended to solve for the coefficients of a sequence generated with input data consisting of an arbitrary starting number and an arbitrary constant differential.